Analyzing the Zeros of the Riemann Zeta Function Using Set-Theoretic and Sweeping Net Methods

1 IntroductionThe Riemann zeta function ζ(s) is a central object in number theory and complex analysis, definedfor complex variables and intimately connected to the distribution of prime numbers through its zeros.The famous Riemann Hypothesis conjectures that all non-trivial zeros of the zeta function lie on thecritical line Re(s) = 12 .In this paper, we explore the Riemann zeta function through the lens of set-theoretic and sweepingnet methods, leveraging creative comparisons of specific sets to gain deeper insight into the distributionof its zeros. By rewording and analyzing the Riemann Hypothesis using set-theoretic arguments, applyingsweeping net techniques, and integrating modal logic interpretations, we aim to provide new perspectivesand support for this profound conjecture.Our objectives are:• Define the zeta function and its properties relevant to the zeros.• Reword the Riemann Hypothesis using set-theoretic language and establish logical equivalence.• Introduce and compare specific sets related to the zeros of ζ(s).• Apply set-theoretic and sweeping net methods to analyze the distribution of zeros.• Provide rigorous proofs about the absence of zeros in certain regions, including mechanical justifi-cations with all steps.• Incorporate modal logic interpretations into the proof.• Discuss implications for the Riemann Hypothesis.